Sunday, October 17, 2010

All for none

And none for all

When I was a teaching assistant during a stint in graduate school, one of my assignments involved an undergraduate abstract algebra class. The professor in charge was keen on teaching (unlike many of his colleagues) and routinely collected and paid attention to student feedback. One day, while he was shuffling through a sheaf of student comments, he burst into laughter and waved a piece of paper at me.

It said, “More small groups!”

“I don't know what it means!” he said. “Do they want me to break them up into small discussion groups more often, or do they want me to spend more time on groups of small order?”

I strongly suspected the former (student groups of four or so) rather than the latter (the two groups of order four—Z4 and Z2 × Z2).

Not all students like having the class broken up into small groups. It lessens one's anonymity to be in a group of three to five rather than in a classroom of thirty to forty. It intrudes on one's ability to blissfully zone out or catch a catnap.

Quite a problem.

It also inspires resistance on the part of some of the best students, who resent carrying the load for their particular group while easy-riding slackers loll about.

But I like the occasional disruption of the standard lecture format with some desk-scooting, semi-organized group babbling, and a bit of cooperative effort. It just refuses—on occasion—to work as intended.

That's a classroom truism, isn't it? As Clausewitz, Moltke, or Powell once said, no battle plan—or lesson plan—survives its first encounter with the enemy. (Not that I mean to characterize my students as the enemy. At least, not usually.)

And what is my (small-group lesson) plan? It has several components:

A dash of variety

A bit of cooperation

A chance to explain to others

A chance to learn from others

A rising tide to lift all the boats

It's that last item that frustratingly doesn't seem to work very well. Most students like the break in routine. Some relish the opportunity to explain things to their peers (especially those who have grasped that explaining something to someone else is about the best possible way to test your understanding of it). Several are eager for assistance from their peers (who are generally less scary than the mean old professor). And some really need it.

These latter are the ones who too often bravely run away. It's a kind of “running in place,” though. They'll scoot their desks into a cluster with their neighbors (although sometimes extra prompting is required). They'll keep their eyes open while worksheets are passed around and they'll fill in their names on their copies. They'll make noncommittal murmuring noises during discussion to indicate their presence.

And at the end of the period they hand in blank worksheets. Or nearly blank.

I'm aware of this, of course, so I roam about the room noting the progress of various groups and individuals and doing a bit of selective prodding. For example, I did this in a calculus class:

“Did you get the same derivative as the other members of your group?”

“Uh, I think so.”

“Perhaps it would be good to check, don't you think?”

“Um, I guess.”

In a louder voice to the whole group:

“Remember that I want the entire group to agree on each step before proceeding to the next, okay? Cross-check each other.”

I wander over to another group.

“Did everyone get the same critical numbers?”

Various responses come back: definite ones from people sitting close together who are working in tandem and less certain ones from those who are hanging back a little. The loners may be totally lost or they may be forging ahead on their own, impatient and unwilling to wait for their less gifted classmates. (Or they just don't play well with others. Sometimes they break out of the pack early, brandishing a filled-out worksheet that they want to be the first to hand in. I send them back to their groups to assist the stragglers, which they don't always appreciate. And sometimes the stragglers don't either!)

“Well, see that you agree and work out any differences.”

I keep moving, trying not to hover.

“All right, guys. How far have you gotten?”

“We're trying to figure out if these are maxes or mins.”

“Okay. Are you agreed on what points you're checking?”

One student leans forward to conceal the blankness of his worksheet. Others are nodding their heads or voicing their agreement. I lean down a bit and speak quietly to the student with the blank paper.

“Your neighbors are on the right track. Copy down their work and ask them for help if you get stuck.”

I gather that some students don't transcribe the joint work of their group because they have nothing to contribute and they don't “own” the results. I appreciate that sensitivity (if indeed that's what it is), but I'm explicitly giving the students permission to help each other across the finish line. On such occasions I routinely say, “All right, everybody! Today is a day for lots of perfect scores! Check with your classmates to verify your work and your answers.” (The subsequent solo exam will show if anything stuck.)

Usually that elicits a bunch of positive responses. One student chortles, “Ten out of ten, here I come!” She has a big smile on her face. But others look worried.

Sure enough, at the end of the period, the boy who hid his paper hasn't finished filling it out. For whatever reason, he kept dawdling instead of piggybacking on his classmates. The girl who was eager for a perfect score looks like she achieved it, transcribing with particular care the detailed consensus solution from her group, making up for some of the difficulties she was having earlier. Most of the class will be keeping company with her at the high end of today's grade distribution. Others will manage to hand in incomplete work or even worksheets on which their incorrect calculations end in hastily substituted correct answers borrowed from classmates. (I read the work, guys. Don't just copy answers!)

Some of my colleagues use group work as the default instructional approach in their classrooms and do their best to avoid the lecture format. It's not, however, a universal solution to the problem of the droning dispensation of knowledge. (Nothing is.) There are as many ways to go wrong in the classroom as there are little knots of students. Since there's no one-size-fits-all solution, all you can do is to keep offering different learning opportunities. You try to do the greatest good for the greatest number, but most of the semester may have slipped by before you discover how to do that with a particular class.

Then a new school term begins and you start all over again with a new batch of students. If you view this as a sequence of constantly renewed opportunities to foster learning and to learn more about teaching, then life is good. If, instead, it puts you in mind of the labors of Sisyphus, you may be in the wrong line of work.

I've been trying to restrain from writing a whole blog post myself on this topic, but will limit myself to a junior-level math course I took while getting my Master's in Math Ed, which I think was called something like Number Systems. The professor started us off with a simplified set of Peano's axioms (from declaring the existence of 1 to induction); by the end of the semester we had worked our way up to imaginary numbers.

The first part of most class hours was professor lecture, then we split into our assigned groups of 4-5 students to work on proving the class's latest theorem. Luckily there were a couple of other fairly bright undergrads in my group, so I didn't have to do all the heavy lifting myself. Also, since I'd already taught school (and had pedagogy coursework while get my teaching credential in California), I knew how to ask "higher-level" questions that probed for understanding, which helped the process along.

P.S. Zeno, I just discovered your email in my spam filter (I keep it set on high, which has the unfortunate side-effect of occasionally screening out emails I actually want). Anyhow, I've approved your email address and forwarded your message to my inbox, where I'll reply at length (oh, no!) when I get a chance. It was good to hear from you so promptly, and I apologize for not checking my spam filter more often or replying sooner.

BTW, did you know that the group Z2 × Z2 is isomorphic -- if that's the correct term; it's been so long I've forgotten -- to the inversion, retrograde and retrograde inversion of a melody (including 12-tone tone-rows)? I wrote a paper on the topic for a Math Ed course.

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